*(English)*Zbl 0727.58026

One studies the codimension two unfolding of an orbit ${{\Gamma}}_{0}$ homoclinic to a saddle-node equilibrium in ${\mathbb{R}}^{d}$. The eigenvalues of the equilibrium are supposed to be positive and negative real parts, except for a simple eigenvalue ${\lambda}_{0}=0$. Several new methods are employed. The bifurcation diagram is obtained by the bifurcation function which is derived by the method of Lyapunov-Schmidt decomposition. The idea of exponential trichotomy is employed to study the linearized equation around ${{\Gamma}}_{0}\xb7$

The method of cross sections and Poincaré mappings are used to study the bifurcation of the periodic orbits from ${{\Gamma}}_{0}\xb7$

Under the Poincaré mapping, submanifolds of certain type, called the u- slices can be found to be fixed and hence bifurcation of periodic orbits is proved.

##### MSC:

37G99 | Local and nonlocal bifurcation theory |

37-99 | Dynamic systems and ergodic theory (MSC2000) |

34C45 | Invariant manifolds (ODE) |

37C70 | Attractors and repellers, topological structure |